[cgmg]

In summary: Just as the complex numbers are defined by adding a new element i such that i^2 = -1, the quaternions are defined by adding elements i, j, k such that i^2 = j^2 = k^2 = ijk = -1.

[akira2501]

Also, see https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...